Modelling of optical logic gates for computer simulation

Modelling of optical logic gates for computer simulation

J.J. Morikuni
D.S. Gao
S.M. Kang

Indexing terms: Logic, Computer simulation

Abstract

The authors have developed a set of equivalent circuit models for optical logic gates based on the saturable gain and saturable absorption phenomena, and have implemented them on the iSMILE circuit simulator. These models have been created with a highly modular design to facilitate the construction of complex logic functions. Several design issues, such as logic threshold and switching speed, have been addressed. Models for the AND, OR, NAND, NOR and NOT gates, are presented, as well as detailed simulation results.

1 Introduction

Optical logic can be performed with lasers through the well-known phenomena of saturable gain and saturable absorption [1-7]. Saturable gain is the phenomenon by which the longitudinal gain of a laser is reduced by lateral injection of photons into its cavity. These photons stimulate emission in a lateral mode which reduces the longitudinal gain and results in a decrease in output power. With this effect, a laser can be turned off optically, a mechanism that can be used to implement optical NOT, OR and NOR functions. Saturable absorption occurs when an absorbing region of a laser is optically pumped by lateral photon injection. The injected photons increase the electron density in the laser cavity which decreases the absorption. With this effect, a laser can be turned on optically, which is a property that can be used to construct optical AND and NAND functions.

Optical logic gates can be controlled either electronically or optically. Electronically controlled logic gates use electrically pumped lasers as the gate inputs, whereas optically controlled logic gate inputs are provided by external light sources. One of the strengths of our models is that they are applicable in both of these environments. That is, they can be used for both purely optical and optoelectronic applications.

We implemented our optical logic models in iSMILE [8], a versatile SPICE-like circuit simulator. Unlike conventional circuit simulators that contain only built-in models, iSMILE allows users to create and define their

Paper 8677J (E10, E13), first received 29th May and in revised form 21st November 1991
The authors are with the Department of Electrical and Computer Engineering and Center for Compound Semiconductor Microelectronics, University of Illinois at Urbana-Champaign, Beckman Institute, 405 N. Mathews Ave., Urbana IL 61801, USA

IEE PROCEEDINGS-J, Vol. 139, No. 2, APRIL 1992
own device models. We have expanded on the multiple quantum-well laser model of Gao et al. [9] to model the saturable gain and absorption effects, and have successfully simulated the AND, OR, NAND, NOR and NOT functions.

2 Implementation of gain saturation

The quantum-well laser model of Gao et al. [9] is based on the well known (single-mode) rate equations for the electron and photon densities $[10,11]$.

$$\begin{aligned} & \frac{d n}{d t}=\frac{J}{q N L_{a}}-B n^{2}-\Gamma g_{m} c^{\prime} S \\ & \frac{d S}{d t}=\beta B n^{2}+\Gamma g_{m} c^{\prime} S-\frac{S}{\tau_{p h}} \end{aligned}$$

Here, $n$ and $S$ represent the average values of the electron and photon densities, respectively; $\Gamma$ is the optical confinement factor; $B$ is the radiative recombination coefficient; $\beta$ is the spontaneous emission coupling coefficient; $L_{a}$ is the quantum-well thickness; $N$ is the number of quantum-wells; and $c^{\prime}$ is the speed of light in the lasing medium. The optical gain term $g_{m}$ is a function which represents carrier injection by a separation of quasiFermi levels [12-15].

$$\begin{aligned} g_{m}= & \frac{4 \pi^{2} q^{2}}{i_{0} m_{e}^{2} c n h E_{p h} m_{e}^{*}+m_{e}^{*}}|M|^{2} \\ & \times\left[f_{c}\left(E_{f v}, E_{c}^{\prime}\right)-f_{c}\left(E_{f v}, E_{c}^{\prime}\right)\right] \end{aligned}$$

This gain function is a very faithful rendition of the true quantum mechanical physics of the device. It was chosen over the conventional form (which assumes that optical gain varies linearly with carrier density) to provide a stronger connection to the physical phenomena of quantum-well lasers [9].

Although very accurate, the $g_{m}$ of eqn. 3 cannot model the saturation of gain that occurs when the photon density $S$ in the cavity becomes large [16, 17]. This saturation effect is the basis of the saturable gain and saturable absorption phenomena, and its characterisation plays a key role in the modelling of optical logic. To incorporate the gain saturation effect into $g_{m}$, the following gain function is implemented:

$$g=\frac{g_{m}}{1+S / S_{a s}}$$

and $g_{m}$ is replaced by $g$ in the rate eqns. 1 and 2 . Once this substitution is made, the optical gain $g$ becomes

dependent not only on the injected carriers, but also on the number of photons in the laser cavity.

The parameter $S_{\text {net }}$ is called the saturation photon density and describes the degree to which the photon density $S$ affects the gain. Physically, if the inverted states of a semiconductor laser are viewed as a two-level system, $S_{\text {net }}$ can be represented by [16]

$$S_{\text {net }}=\frac{2 \pi n^{2}}{\lambda^{2} c} \frac{\tau_{c}}{\tau_{c u t}^{2}}$$

where $\tau_{c u t}$ is the intraband carrier relaxation time, and $\tau_{c}$ is the carrier lifetime.

Fig. 1 shows the equivalent circuit model of the quantum-well laser with gain saturation taken into
the output power is modelled by a linear dependent voltage source.

Fig. 2 shows the DC light-current response for this model. The laser used in this simulation had dimensions of $L=1000 \mu \mathrm{~m}, W=3 \mu \mathrm{~m}$. Results for this simulation compare favourably with data obtained from McDonnell Douglas [18]. iSMILE predicts a threshold current of 16.9 mA with an $\mathrm{L} / \mathrm{I}$ slope of $0.44 \mathrm{~mW} / \mathrm{mA}$, whereas actual data show a threshold current of 17 mA and an $\mathrm{L} / \mathrm{I}$ slope of $0.45 \mathrm{~mW} / \mathrm{mA}$.

At this point, it is important to mention some of the assumptions that were made in the creation of the quantum-well laser model [9]. Some of the more important ones are the assumption of uniform plane waves

Fig. 1 Quantum-well laser model
account. There are three node voltages in this model. The first, $V_{0}$, represents the electrical characteristics of the laser. The similarity of the laser diode I/V characteristic below threshold to that of an ordinary diode suggests that they have similar nonradiative recombination mechanisms [9]. Thus, the nonradiative recombination current of the laser diode is modelled as an ideal diode $D$ in parallel with a capacitor, $C_{d}$. Nonlinear resistors model the recombination currents $I_{x p}$ and $I_{x r}$, due to spontaneous and stimulated emission, respectively, and $C_{b}$ represents the storage effect of charge in the active region.

The second node $V_{0}$ represents the optical characteristics of the laser. The value of $V_{0}$ is related to the photon density in the cavity by

$$V_{0}=\frac{S}{S_{\text {net }}}$$

The dependent current sources attached to this optical node represent the contributions of spontaneous and stimulated emission to the photon density, and the resistor is used to take photon loss in the cavity into account. The ‘optical capacitance’ models the storage effects of photons in the laser cavity. Finally, the third node $P_{\text {out }}$ represents the output power of the laser. Output power and photon density are linearly related, so

Fig. 2 DC response: single laser DC L/I curve
within the laser cavity, the use of average values for the photon densities, uniform pumping, neglecting of light holes, and assignment of the difference in quasi-Fermi levels to the voltage applied across the laser. It should also be emphasised that the saturable gain and absorption models make the further assumption that radiation into nonlasing modes is negligible.

3 Saturable gain model

3.1 Equivalent circuit

The saturable gain phenomenon, which is also known as gain quenching, occurs when radiation is incident on a laser cavity, causing the inverted population in the laser to be exhausted, or quenched [19-21]. The degree to which the laser is quenched depends on the strength of the radiation. One way to realise this effect is to use the configuration of Fig. 3, in which the incident radiation is supplied by a control laser. In this schematic representation, the shaded regions indicate areas of electrical contact (metal), and the arrows indicate the flow of outgoing radiation (photons).

Fig. 3 Structure for realising saturable gain
If the output laser is turned on with the control laser off, power will emerge from the output cavity. However, if the control laser is turned on, control laser photons will be injected into the output cavity reducing the output gain. The end result is a reduction in the amount of

power emitted by the output laser, making it possible to turn off a laser (or at least reduce its output) using another laser.

Fig. 4 is a photograph of an actual layout of a saturable gain test device [21], which consists of two identical

Fig. 4 Saturable gain test device
devices, one on the left side of the chip and one on the right side. The four dark areas are $12 \mu \mathrm{~m}$ thick layers of gold that serve as electrical contacts. The white borders are made of a titanium/platinum/gold alloy which enhances conductivity. The dark lines between the white borders serve as electrical isolation between the control and output lasers and have a resistance of approximately $10-12 \mathrm{k} \Omega$. The dimensions of the output laser are $L=500 \mu \mathrm{~m}, W=4 \mu \mathrm{~m}$, and the dimensions of the control laser are $L=200 \mu \mathrm{~m}, W=32 \mu \mathrm{~m}$. The rectangular region on the output laser gold contact is an etched facet.

The gain function of eqn. 4 must now be modified to include the effect of photons from the control laser. The contribution of control laser photons is referred to as $S_{\text {extra }}$ and is incorporated into the original gain function of eqn. 4 as

$$g=\frac{g_{\mathrm{m}}}{1+\frac{S+S_{\text {extra }}}{S_{\text {sat }}}}$$

Turning on the control laser will increase $S_{\text {extra }}$ which will, in turn, decrease the gain of the output laser. An important point to realise is that the quantity $S_{\text {extra }}$ is not the total photon density in the control laser cavity; rather, it is only that fraction of the control laser photons that exits the cavity and enters the output laser. That is, $S_{\text {extra }}$ represents only that fraction of control laser photons that is injected into the cross-hatched, overlap area of Fig. 5.

An expression for this fraction has been derived in the Appendix and is stated here as

$$S_{\text {extra }}=K S_{\text {control }}$$

where $S_{\text {control }}$ is the total photon density in the control laser, and the multiplicative overlap factor is given by

$$K=\frac{k_{0}}{4} \frac{W_{1}}{L_{1}} \ln \left(\frac{1}{R_{2}}\right) \frac{e^{g W_{1}}-1}{g W_{1}}\left(1+R_{e} e^{g W_{1}}\right)\left(1-R_{e}\right)$$

Here, $W_{1}$ is the width of the output laser; $W_{2}$ is the width of the control laser; $L_{1}$ is the length of the output laser; $R_{e}$ is the reflectivity of the interface between the control and output lasers; $R_{2}$ is the reflectivity of the control laser facet; and $R_{e}$ is the reflectivity of etched facet. An

Fig. 5 Illustration of overlap region
important point to realise about eqn. 9 is that $S_{\text {extra }}$ is averaged over the entire output laser cavity (see the Appendix).

As the only output node of the equivalent circuit of Fig. 1 is the output power node, another output node representing $S_{\text {extra }}$ has been added to the model. As $S_{\text {extra }}$ is linearly related to the photon density by eqn. 8 , this is accomplished with a linear dependent voltage source. To model the saturable gain effect of eqn. 7, the equivalent circuit of Fig. 1 has been further modified by adding an $S_{\text {extra }}$ input node. This is done by providing an external node across which a voltage representing $S_{\text {extra }}$ can be placed. This node voltage is referred to as $V_{A}$. Although there is no current flow into this node, a resistor $R_{e}$ of arbitrary value is placed across $V_{A}$ to provide a DC path. The modified equivalent circuit model is depicted in Fig. 6 .

Fig. 6 Saturable gain equivalent circuit model
As we have modified the laser model so that it can both send and receive a flux of photons ( $S_{\text {extra }}$ ), both control and output laser models can be created from the same model. Fig. 7 depicts the saturable gain model of Fig. 6 as a black box ‘universal gain element’, an element with which all devices that are necessary for modelling the saturable gain phenomenon can be constructed.

The universal gain element can, for example, be used as a control laser by grounding the $S_{\text {extra, la }}$ input and igonoring the $P_{\text {out }}$ output. An output laser would be modelled by ignoring the $S_{\text {extra, out }}$ output. By providing this type of modularity, it is possible to construct complex logic gates using only one ‘fundamental building block’. In this manner, it is not necessary to create a new model for every new gate or configuration. It is sufficient

simply to modify the iSMILE input deck to include one or more of these modules. As an analogy, it is helpful to think of the universal gain element in an optical logic circuit in the same way as a transistor in an electronic

Fig. 7 Universal gain element
logic circuit. One does not make a new model for every type of electronic logic gate in, for example, SPICE. Rather, one connects several transistors together at the circuit, or input deck level. Note, in passing, that it takes two electronic transistors to model a CMOS logic gate compared to one active junction to model a given optical logic function [18].

3.2 Optical logic gates based on saturable gain

With the universal gain element, it is possible to create models for three types of logic function NOT, NOR and OR. All three of these functions are based on the saturable gain phenomenon. That is, all three functions operate by turning a laser off with another laser or lasers.

3.2.1 Optical NOT function

An optical NOT gate would have the structure shown in Figs. 8 and 9. The output laser is always turned on, so

Fig. 8 Optical inverter

Fig. 9 NOT gate schematic diagram
there is always a stream of carriers flowing through the contact to maintain a population inversion. Thus, with the control laser initially off, the output of the output laser would be ‘high’. When the control laser is turned on, photons from the control laser bombard the output cavity, thereby reducing the output gain. This results in a drop in the output power, signifying a logic ‘low’.

The optical inverter can be modelled with the universal gain element, as depicted in Fig. 10. The $S_{\text {syste }}$ input is grounded for the control laser, a constant bias is kept across the output laser, and a variable input is applied across the control laser. The $S_{\text {syste }}$ output on the output laser and the $P_{\text {out }}$ output on the control laser are ignored.

It is to be emphasised that they are available; for this particular application, however, they are not of interest.

A sample simulation of the optical inverter is shown in Fig. 11. This graph depicts an optical inverter with

Fig. 10 Optical NOT, circuit level representation

Fig. 11 Optical NOT simulation
output power
control laser pump
output laser dimensions $L=500 \mu \mathrm{~m}, W=3 \mu \mathrm{~m}$ and control laser dimensions $L=200 \mu \mathrm{~m}, W=30 \mu \mathrm{~m}$. As can be deduced from Fig. 5, the size of the overlap region is $30 \mu \mathrm{~m}$ by $3 \mu \mathrm{~m}$. Ideally, a large swing between the output power in the high state ( $P_{\text {high }}$ ) and the output power in the low state ( $P_{\text {low }}$ ) is desirable. However, this simulation showed a drop of only about $5.6 \%$. The reason for this is the relatively small area of interaction between the control and the output lasers. Examination of the optical inverter (Fig. 5) will show that the effective overlap region is approximately the width of the control laser times the width of the output laser (the crosshatched region). Increasing this overlap area will result in larger power swings. Logically, this can be seen by realising that more overlap results in more control laser photons entering the output cavity and, hence, larger drop in output gain.

The simulation of Fig. 11 appears to have a very slow response time. That is, there appears to be a considerable lag between the time the control laser pump turns on and the time the output laser saturates. The apparent time lag arises from the fact that once the control laser pump voltage turns on, the control laser itself takes time to turn on. Once the control laser turns on, the output gain takes time to saturate. After this happens, the output power finally decreases. The time lag is due to all of the events involved, not just the output switching.

If a stream of photons is used instead of a control laser, the switching time would be much faster. This would represent a true optically controlled logic gate. As the previous examples required an electrical bias to be supplied to turn on the control laser, they were, in effect, electronically controlled optical logic gates. As the extra photon density is represented by a node voltage, it is possible to investigate the properties of optically controlled

optical switching by replacing the control laser model with an ideal voltage source. The results of such a simulation are shown in Fig. 12. This figure shows an actual switching time in the order of tens of picoseconds. The dominating parameter for the switching time in this model is the capacitor on the optical node (Fig. 1) that models the photon storage effects in the laser cavity [9].

Fig. 12 Switching via photon injection
$\qquad$ output power
isjected photons

3.2.2 Optical NOR function

The logical NOR function can also be modelled with saturable gain. In the case of the NOT function, the gain of the output laser was saturated, or reduced, by the injection of photons by a control laser. To model the NOR function, all that has to be done is to add another control laser (Figs. 13 and 14) and allow the output gain

Fig. 13 Optical NOR gate

Fig. 14 NOR gate schematic diagram
to depend on either (or both) of the control lasers. That is, the output gain must be allowed to saturate if either of the control lasers turns on. If both of them are on, then the output gain should saturate even further. This can be understood by referring to eqns. 7 and 8 . When one control laser is turned on, $S_{\text {extra }}$ becomes nonzero, and the gain decreases. When the second control laser turns on, $S_{\text {extra }}$ becomes even larger, causing the output gain to decrease even further.

With the universal gain element, it is very easy to model the NOR function by considering the effect of

IEE PROCEEDINGS-J, Vol. 139, No. 2, APRIL 1992
both overlap regions jointly. Instead of placing two different $S_{\text {extra }}$ inputs on the output laser, the two control laser $S_{\text {extra }}$ outputs are added, and the sum is passed to the output laser. This approach is consistent with the original quantum-well laser model which considers only the average values of physical quantities such as photon density [9]. Fig. 15 illustrates this point. In addition to receiving the sum of the $S_{\text {extra }}$ inputs, the output laser must also be informed of the total area of the overlap regions.

Fig. 15 Optical NOR function, circuit level representation
The advantage of this method is that it is very easy to create new gate configurations. As an example of the flexibility this approach affords, to create a three-input NOR gate, all that would have to be done is to add another control laser and dependent voltage source in Fig. 15 and to increase the effective (total) overlap area in the iSMILE input deck for the output laser. Note that it is entirely possible to build a new model for the optical NOR function that has two overlap regions and takes both into account separately, but this would be impractical. If this approach were taken, a different model would have to be created for every possible type of gate.

The results of an optical NOR simulation with the same dimensions as the inverter in Fig. 11 (identical control laser) are shown in Fig. 16. At 1 ns , the output laser turns on and stays on for the duration of the simulation. When the first control laser turns on, the output power drops, signifying a logic low. When the second control laser turns on, the output power drops even further. As the output is already low, this has no effect in the Boolean sense. However, our model successfully illustrates the effect on the output gain of having two control

Fig. 16 Optical NOR simulation
output power
control laser A pump
control laser B pump

lasers on simultaneously. At 20 ns , the first control laser turns off, leaving the second one on. As expected, the output power rises to the level that it was at with only one control laser. The output power rises, but it is still a logic zero. It is important to realise that this is the same logic level, even though a different control laser is on. This is important because the contribution of a control

Fig. 17 Optical $O R$

Fig. 18 OR gate schematic diagram

Fig. 19 Optical OR simulation
$\qquad$ output power control laser A pump control laser B pump
laser to the output power should be the same, no matter which control laser is on. At 25 ns , the second control laser turns off, leaving the output laser in its original high state.

3.2.3 Optical OR function

The optical OR function can be modelled as an optical NOR function with an inverter at the end (Figs. 17 and 18). Physically, the operation is identical to that of the NOR gate, except that the output is inverted. Fig. 19 depicts an optical OR simulation. Fig. 20 is the circuit level representation of the optical OR function.

4 Saturable absorption model

4.1 Equivalent circuit

If a laser cavity is not pumped, there is no injection of carriers, no population inversion and, hence, no lasing. Instead of triggering stimulated emission, any light incident on the cavity under zero current or voltage bias is absorbed. When this is done, carriers can no longer be injected, and no inversion can occur. Consider the configuration depicted in Fig. 21. The control laser is fully pumped. A region of the output laser indicated by the unshaded portion, however, has no metal contact, and hence, is not pumped. As described above, because no carriers can enter this region, no pumping or population inversion can occur. The region will be absorbing. Thus, even if voltage or current is applied to the metal regions of the output laser, lasing will not occur.

Fig. 22 is a photograph of a layout of a saturable absorption test device [21]. As with the saturable gain layout, the depicted layout consists of two identical devices, one on the left side of the chip and one on the right side. The various layers are identical to those of the saturable gain test device shown in Fig. 4. The dimensions of the output laser in Fig. 22 are $L=500 \mu \mathrm{~m}$, $W=4 \mu \mathrm{~m}$, and the dimensions of the control laser are $L=200 \mu \mathrm{~m}, W=8 \mu \mathrm{~m}$. Notice the white area halfway down the output laser. This is the unpumped region that gives the output laser its saturable absorption characteristic. On the saturable gain test chip, this white area was covered with metal.

Although the absorption mechanism can be thought of as the inverse mechanism of gain, mathematically they both take on the same form. Examination of the gain/ voltage characteristic of the quantum-well laser model (Fig. 23) [9] shows that, when the applied voltage is less

Fig. 20 Optical OR: circuit level representation

than some threshold voltage, the gain is negative. By definition, because absorption is that region of the gain function that is negative, eqn. 4 can be written in the form of

$$g=\frac{g_{\mathrm{m}}}{1+\frac{S}{S_{\mathrm{sat}}}}=\begin{gathered} \text { absorption } \\ \text { gain } \\ V_{\mathrm{in}}<V_{\mathrm{th}} \end{gathered}$$

where $V_{\text {in }}$ is the threshold voltage required to turn on the laser.

Fig. 21 Structure for realising saturable absorption

Fig. 22 Saturable absorption test device

Fig. 23 Gain/voltage characteristic
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The saturable absorption phenomenon is analagous to the saturable gain effect. Another way to express the absorption $a$ is

$$a=\frac{g_{\mathrm{m}}}{1+\frac{S+S_{\text {extra }}}{S_{\text {sat }}}}\left(V_{\text {in }}<V_{\text {th }}\right)$$

As seen in Fig. 21, $V_{\text {in }}$ for the nonmetalised region is zero. It is easy to see from eqn. 11 and Fig. 23 that the absorption, which is a negative number, will decrease in magnitude as the strength of the control laser ( $S_{\text {extra }}$ ) increases. That is, the stronger $S_{\text {extra }}$ is, the closer the absorption will go to zero. When the absorption becomes very small, light in the output cavity will be able to pass through the unpumped region instead of being absorbed.

To implement the saturable absorption phenomenon, the gain function was modified to account for the lack of electrical pumping in the absorbing region. This was not difficult since absorption takes the same form mathematically as gain (eqn. 11). The challenge was in describing the interaction between the absorbing region and the rest of the output laser cavity. Since the quantum-well laser model only models the average photon density in the laser cavity [9], it is not well suited for modelling localised phenomena such as bombardment of a region of the cavity by photons. Unlike the saturable gain phenomenon, in which the extra photon density could be averaged over the entire output cavity, for the saturable absorption phenomenon this does not make physical sense. The control laser saturates the absorption in the unpumped region and, as such, $S_{\text {extra }}$ should only affect the absorbing region. The solution chosen is two fold. The first step is to recognise that absorption is a form of loss. The net gain in any laser cavity is taken to be the gain in the cavity minus the losses. Usually, these losses take the form of mirror reflectivities, imperfect facets etc., but these losses can also include any absorbing elements. The second step is to determine how to relate the absorption as loss to the rest of the pumped laser cavity. As a first-degree approximation, the net gain in the output laser is considered to be the average of the weighted sum of the gain in the pumped region and the absorption in the unpumped region. If the weighting is estimated to be linear, the net gain in the laser can be approximated as

$$\operatorname{gain}_{\text {net }}=\frac{\operatorname{gain}_{\text {pumped }} A_{\text {pumped }}+\text { absorption }_{\text {unpumped }} A_{\text {unpumped }}}{A_{\text {pumped }}+A_{\text {unpumped }}}$$

In this expression, $A_{\text {pumped }}$ and $A_{\text {unpumped }}$ represent the areas of the pumped and unpumped regions of the output laser, respectively; gain ${ }_{\text {pumped }}$ represents the gain in the pumped regions; and absorption ${ }_{\text {unpumped }}$ represents the absorption in the unpumped region. Since the photon interaction from the control laser does not (to a first degree) affect the pumped regions of the output laser, the gain in the pumped regions assumes the form of eqn. 13, and the absorption in the unpumped region assumes the form of eqn. 14.

$$\begin{aligned} & \operatorname{gain}_{\text {pumped }}=\frac{g_{\mathrm{m}}}{1+\frac{S}{S_{\text {sat }}}} \\ & \text { absorption }_{\text {unpumped }}=\frac{g_{\mathrm{m}}}{1+\frac{S+S_{\text {extra }}}{S_{\text {sat }}}} \end{aligned}$$

In this manner, some degree of photon localisation is obtained while the average value approach used in the quantum-well laser model is retained.

There was one modification that had to be made to the gain expressions of eqns. 12-14, which stems from the fact that the photon density in the output cavity $S$ is not uniform throughout the output cavity. Recall that for the saturable gain case, the photon density was assumed to be approximately uniform over the entire cavity. For saturable absorption, such an assumption is unrealistic. The photon density in the absorbing region is significantly less than the average photon density. Similarly, $S$ in the gain regions is greater than the average photon density. These two facts can be represented by introducing the density localisation factors $\sigma_{1}$ and $\sigma_{2}$ :

$$\begin{aligned} & S_{\text {unpumped }}=\sigma_{1} S \\ & S_{\text {pumped }}=\sigma_{2} S \end{aligned}$$

Here, $S$ is the average photon density and $\sigma_{1}<1$ and $\sigma_{2}>1$. Invoking another relation,

$$A_{\text {total }} S=A_{\text {pumped }} S_{\text {pumped }}+A_{\text {unpumped }} S_{\text {unpumped }}$$

$\sigma_{2}$ can be determined in terms of $\sigma_{1}$.

$$\sigma_{2}=\frac{A_{\text {total }}-\sigma_{1} A_{\text {unpumped }}}{A_{\text {pumped }}}$$

In eqn. $18, \sigma_{1}$ is adjusted to fit experimental data; i.e. $\sigma_{1}$ is user specified. This representation is not completely accurate as $\sigma_{1}$ (and hence $\sigma_{2}$ ) should be allowed to vary with the degree of absorption. As a first-order approximation, however, good simulation results have been obtained with this model. The complete gain function describing the output laser is, then, eqn. 12 with the following modifications:

$$\begin{aligned} & \text { gain }_{\text {pumped }}=\frac{g_{m}}{1+\frac{\sigma_{2} S}{S_{\text {sat }}}} \\ & \text { absorption }_{\text {unpumped }}=\frac{g_{m}}{1+\frac{\sigma_{1} S+S_{\text {extra }}}{S_{\text {sat }}}} \end{aligned}$$

It is this gain function that is substituted for $g_{m}$ in the rate eqns. 1 and 2 to model saturable absorption.

4.2 Optical logic gates based on saturable absorption

We have demonstrated that, through the saturable absorption phenomenon, it is possible to turn on a laser using another laser. Logic gates that depend on one or any combination of their inputs to be on can be modelled using this mechanism. The only differences between the saturable gain and absorption equivalent circuits are the gain functions and the user definable parameters. The equivalent circuit topology is unchanged. It was, however, not possible to define a ‘universal absorption element’ as in the case of saturable gain. For a saturable gain circuit of more than one input, it was possible to model the net effect of all the inputs together. Physically, this was reasonable as each input had the same effect on the output laser as the others did. However, for saturable absorption, each absorbing region must be considered separately. To see why this is so, try grouping the absorbing regions together into one big effective region as was done for saturable gain. If this is done, it becomes possible to turn on the output laser with only one of the control lasers if that control laser is strong enough. In reality, the
output laser cannot be activated by only one input no matter how strong it is, up to a certain, reasonable point. For this reason, a separate model must be created for each different saturable absorption device desired. This was not difficult, however, as only a few specific lines of code needed to be modified.

4.2.1 Optical buffer

As an analogue to the optical inverter, the circuit for an optical buffer is presented in Figs. 24 and 25. This circuit demonstrates the overall behaviour of the saturable absorption mechanism and provides a basic understanding of how to construct more complex functions.

Fig. 24 Optical buffer

Fig. 25 Buffer schematic diagram
A sample simulation of the optical buffer is shown in Fig. 26. This graph is a simulation of a device that has control laser dimensions $L=200 \mu \mathrm{~m}, W=30 \mu \mathrm{~m}$ and output laser dimensions $L=500 \mu \mathrm{~m}, W=3 \mu \mathrm{~m}$. As can be deduced from the structure shown in Fig. 24, the size of the unpumped region is $30 \mu \mathrm{~m}$ by $3 \mu \mathrm{~m}$. From Fig. 26, it is apparent that the output laser will not turn on until the control laser is activated.

Fig. 26 Optical buffer simulation
output power
output laser bias voltage
control laser pump voltage

4.2.2 Optical AND function

The logical AND operation can be realised with saturable absorption. To model the AND function, an additional unpumped region must be added to the output

laser as well as an additional control laser to pump it. Schematic diagrams are shown in Figs. 27 and 28, and simulation results are presented in Fig. 29.

Fig. 27 Optical $A N D$

Fig. 28 AND gate schematic diagram

Fig. 29 Optical AND simulation
$\qquad$ output power
$\qquad$ control laser A pump
$\qquad$ control laser B pump

Fig. 30 Optical $N A N D$

Fig. 31 NAND gate schematic diagram
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4.2.3 Optical NAND function

The NAND function can be performed optically by placing an inverter at the output of the optical AND gate. As seen in the schematic diagrams of Figs. 30 and 31, the NAND function utilises both saturable gain and saturable absorption principles. A simulation is depicted in Fig. 32.

Fig. 32 Optical NAND simulation
$\qquad$ output power
$\qquad$ control laser A pump
$\qquad$ control laser B pump

5 Design issues

In addition to being a valuable simulation and analysis tool, the saturable gain and absorption models can provide insight into various design issues. Several of these issues are addressed here.

5.1 Saturable gain: the overlap problem

The main problem that exists with the saturable gain logic gates presented is the small change in output power that defines the logic threshold. Ideally, the difference in output power between high and low states should be as wide as possible. The ideal value for a logic zero would be 0 mW , and the ideal value for a logic one would be the laser’s maximum output power. However, we have shown that this is not possible. The problem lies in the fact that, for saturable gain devices, the gain in the output device is saturated, not eliminated. That is, when the control laser turns on, the output gain does not go to zero, it is merely reduced. Eqn. 7 showed that the drop in output gain varies directly with the magnitude of $S_{\text {extra }}$. As the strength of the control laser increases, $S_{\text {extra }}$ increases, and, consequently, the drop in output gain increases. However, the derivation of $S_{\text {extra }}$ shows that $S_{\text {extra }}$ is only a fraction of the control laser’s average photon density. Thus, in general, $S_{\text {extra }}$ will be less than $S$ of the output laser in eqn. 7. How much less is determined by the dimensions of the lasers etc., as illustrated in eqn. 9.

Of course, it is possible to make $S_{\text {extra }}$ large by increasing the size of the control laser, but there are always the practical constraints of heat dissipation and area. These problems have been encountered with the test chips presented in Figs. 4 and 22. In particular, to attain strong enough power from the control lasers to drive the output lasers, pump currents in the hundreds of milliamps were required. In some instances, it was necessary to switched to pulsed operation to avoid melting the chip. Despite these drawbacks, because the size of the overlap area is the most easily controlled parameter in eqn. 9 , scaling device sizes seems to be the only solution at this point.

Consider, as an exercise, the case in which the control laser is sufficiently strong to produce an extra photon

density that is comparable to the average photon density in the output laser. Then, making the following assumption:

$$\frac{S+S_{\text {expo }}}{S_{\text {tot }}} \gg 1$$

the output gain drops only by a factor of two. Numerical evaluation of eqn. 9 for a few sample cases will show that it is difficult to make $S_{\text {expo }}$ even this large.

This, then, shows the limitations of using the simple cross-coupled laser configuration for saturable gain. Because the overlap area seems to be the dominant factor in increasing $S_{\text {expo }}$, the most obvious solution would be to develop a new geometry that created a higher degree of overlap [21]. Indeed, current work at McDonnell Douglas is pointed in this direction.

5.2 New geometries

The next issue to address is how to modify the simple model developed here to handle these types of new geometry. The Appendix gives a detailed derivation of the extra photon density based on a simple rectangular overlap region. As discussed at the end of the Appendix to handle new types of geometry, this portion of the model will have to be modified. Alternatively, the curve fitting parameter $k_{g}$ can be used to calibrate the model to the new geometry. In this way, the model achieves a large degree of flexibility with only a minimal amount of modification.

As an example of how the $k_{g}$ parameter can be used to model a gate with larger overlap, consider an inverter of some ‘new’ geometry that allows such an overlap (Fig. 33). This inverter has a very big drop in output power corresponding to the activation of the control laser. Thus, when the technology or method to achieve large overlap becomes available, better performance will become possible.

Fig. 33 Simulation of inverter with large overlap area output power pump voltage

5.3 Mode spectrum of output laser

It is generally recognised that the mode spectrum of a laser experiences some degree of linewidth broadening due to lateral modes [22], an effect which should be small under normal operating conditions [23]. However, injection of light by the control laser produces a large amount of stimulated emission in the lateral direction of the output laser [21]. This large amount could theoretically be enough to broaden the spectrum of the output laser by a nonnegligible amount. Our models have been developed for circuit-level simulation, however, and, as
such, do not account for detailed effects such as this. An involved analysis encompassing this phenomenon is better handled by a device-level simulator, such as MINILASE [24].

6 Summary

We have developed equivalent circuit models for bistable optoelectronic devices based on the saturable gain and saturable absorption phenomena and have implemented them into the iSMILE circuit simulator. In this paper, we presented several examples and simulations of these models, including the AND, OR, NAND, NOR, and NOT gates.

The simulations that we presented demonstrated both the advantages and disadvantages of implementing optical logic with saturable gain/absorption. Representative of the advantages is the extremely fast switching times achievable. Simulation results suggest that switching speeds in the tens of picoseconds should be attainable. Among the disadvantages of implementing optical logic with saturable gain/absorption, the most obvious is the inability to produce wide logic thresholds. This is, perhaps, the most serious problem as, without a sizable spread in logic threshold, noise margins become very small, jeopardising signal integrity. Our models successfully reflect this low threshold; furthermore, based on our analysis, we have been able to suggest ways to avoid this problem. The most obvious solution is to increase the effective overlap area in the active region.

The optical logic gates simulated in this research were actually electronically controlled. The control/input photons that initiated optical switching were produced by electrically pumped lasers. One of the strengths of our models is their ability to simulate purely optical switching events; that is, events whose control inputs are optically activated. The simulation of Fig. 12 demonstrated this capability.

We have demonstrated both the feasibility and the desirability of using saturable gain/absorption based optical logic. The models created in this research have served not only as numerical tools with which to analyse these gates, but they have also served to accentuate various nontrivial design issues. Although we have pointed out several of the shortcomings of saturable gain/ absorption based optical logic gates, we also demonstrated their potential for high-speed digital logic circuits.

7 Acknowledgments

The authors wish to thank J.A. Priest and C.L. Balestra of the McDonnell Douglas Corporation for detailing the work that their company has done with saturable gain/ absorption based optical logic gates, and Y. Leblebici for his technical discussions and critical comments.

B References

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5 NATHAN, M.I., MARINACE, J.C., RUTZ, R.F., MICHEL, A.E., and LASHER, G.J.: ‘GaAs injection laser with novel mode control and switching properties’, J. Appl. Phys., 1965, 36, (2), pp. 473-480
6 TIPPETT, J., BERKOWITZ, D., CLAPP, L., KOESTER, C., and VANDERBURGH, A.: ‘Optical and electrooptical information processing’ (The Massachusetts Institute of Technology Press, Cambridge, MA, 1965)
7 LASHER, G.J.: ‘Analysis of a proposed bistable injection laser’, Solid-State Electron., 1964, 7, pp. 707-716
8 YANG, A.T., and KANG, S.M.: ‘ISMILE: a novel circuit simulation program with emphasis on new device model development’. Proc. 26th ACM/IEEE Design Automation Conf., 1989, pp. 630633
9 GAO, D.S., KANG, S.M., BRYAN, R.P., and COLEMAN, J.I.: ‘Modeling of quantum-well lasers for computer-aided analysis of optoelectronic integrated circuits’, IEEE J. Quantum Electron., 1990, 26, (7), pp. 1206-1216
10 ARAKAWA, Y., and YARIV, A.: ‘Quantum well lasers - gain, spectra, dynamics’, IEEE J. Quantum Electron., 1986, QE-22, (9), pp. 1887-1899
11 YEE, T.K., and WELFORD, D.: ‘A multimodel rate-equation analysis for semiconductor lasers applied to direct intensity modulation of individual longitudinal modes’, IEEE J. Quantum Electron., 1986, QE-22, (11), pp. 2116-2122
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13 DUTTA, N.K.: ‘Temperature dependence of threshold current of GaAs quantum well lasers’, Electron. Lett., 1982, 18, (11), pp. 451453
14 ARAKAWA, Y., and YARIV, A.: ‘Theory of gain, modulation response, and spectral line width in AlGaAs quantum well lasers’, IEEE J. Quantum Electron., 1985, QE-21, (10), pp. 1666-1674
15 CASEY, H.C., and PANISH, M.B.: ‘Heterostructure lasers’ (Academic Press, New York, 1978)
16 CHANNIN, D.J.: ‘Effect of gain saturation on injection laser switching’, J. Appl. Phys., 1979, 50, (6), pp. 3838-3860
17 BOURKOFF, E., and LIU, X.Y.: ‘Deep-level trap model of diode laser modulation: significance of spontaneous emission and gain saturation’, J. Appl. Phys., 1989, 65, (8), pp. 2912-2917
18 BALESTRA, C., CLAY, B., and PRIEST, J.A.: ‘Photonics technology’. McDonnell Douglas Internal Memorandum
19 HALL, R.N., FENNER, G.E., KINGSLEY, J.P., SOLTYS, T.J., and CARLSON, R.O.: ‘Coherent light emission from GaAs junctions’, Phys. Rev. Lett., 1962, 9, (9), pp. 366-368
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21 BALESTRA, C., and PRIEST, J.A.: McDonnell Douglas Internal Memorandum
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24 SONG, G.H., HESS, K., KERKHOVEN, T., and RAVAIOLI, U.: ‘Two dimensional simulation of quantum well lasers’, Eur. Trans. Telecommun. Relat. Technol., 1990, 1, (4), pp. 375-381
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9 Appendix: Analysis of photons in the overlap region

The overlap region of the cross-coupled optical cavities is depicted in Fig. 34.

The number of extra photons in the output cavity that have been contributed by the control laser can be approximated by extending the analysis presented in Reference 9. If the overlap region is considered (as a firstdegree approximation) to be a Fabry-Pérot resonator, and if the uniform plane wave approximation is used, the differential equations describing the light intensity in this

IEE PROCEEDINGS-J, Vol. 139, No. 2, APRIL 1992
region are $[25,26]$

$$\frac{d I^{+}(z)}{d z}=g I^{+}(z)+r_{s p} E_{p h}$$

$$\frac{d I^{-}(z)}{d z}=-g I^{+}(z)-r_{s p} E_{p h}$$

where

$$r_{s p}=\frac{\beta B n^{2}}{2}$$

Next, the appropriate boundary conditions must be applied to the cavity:

$$\begin{gathered} I^{+}(0)=R_{s} I^{-}(0) \\ I^{-}\left(W_{1}\right)=R_{s} I^{+}\left(W_{1}\right) \end{gathered}$$

This results in the following solutions for the forward and backward travelling waves $I^{+}$and $I^{-}$:

$$\begin{gathered} I^{+}(z)=\frac{r_{s p} E_{p h}}{g}\left[\frac{\left(1+R_{s} e^{e W t}\right)-R_{s}\left(1+R_{s} e^{e W t}\right)}{1-R_{s} R_{s} e^{2 e W t}}\right. \\ \left.\times e^{g z}-1\right] \\ I^{-}(z)=\frac{r_{s p} E_{p h}}{g}\left[\frac{\left(1+R_{s} e^{e W t}\right)-R_{s}\left(1+R_{s} e^{e W t}\right)}{1-R_{s} R_{s} e^{2 e W t}}\right. \\ \left.\times e^{g\left(W_{1}-z\right)}-1\right] \end{gathered}$$

After some algebraic manipulations

$$\begin{aligned} & I^{+}(z)=I^{+}(0) e^{g z}+\frac{r_{s p} E_{p h}}{g}[(e^{g z}-1)] \\ & I^{-}(z)=I^{-}\left(W_{1}\right) e^{g\left(W_{1}-z\right)}+\frac{r_{s p} E_{p h}}{g}\left[e^{g\left(W_{1}-z\right)}-1\right] \end{aligned}$$

where the first term of each sum can be considered as a ‘small signal’ part and the second term as an additive

Fig. 34 Output laser overlap region
part. Neglecting the additive part in this analysis and taking into account the fact that $I^{-}\left(W_{1}\right)=R_{s} I^{+}\left(W_{1}\right)$,

$$I^{-}(z)=\left[R_{s} I^{+}(0) e^{g W_{1}}\right] e^{g\left(W_{1}-z\right)}$$

Realising that the output power and the average photon density in the control laser are related by

$$P_{\text {out, control }}=\frac{W_{2} N L_{s} E_{p h} C}{4} \ln \left(\frac{1}{R_{2}}\right) \delta_{\text {control }}$$

and that

$$I^{+}(0)=\left(1-R_{\mathrm{a}}\right) \frac{P_{\text {out }, \text { control }}}{W_{2} N L_{\mathrm{a}}}$$

the intensity of the right-going travelling wave at $z=0$ can be written as

$$I^{+}(0)=\left(\frac{1-R_{\mathrm{a}}}{4}\right) E_{\mathrm{ph}} c^{\prime} \ln \left(\frac{1}{R_{2}}\right) S_{\text {control }}$$

The next step is to calculate the photon density in the overlap region that is due to the control laser:

$$S_{\text {additional }}=\frac{1}{E_{\mathrm{ph}} c^{\prime} W_{1}} \int_{0}^{\boldsymbol{W}_{1}}\left[I^{+}(z)+I^{-}(z)\right] d z$$

where $c^{\prime}$ is the speed of light in the lasing medium and $E_{\text {ph }}$ is the photon energy. Using all of the above relations, this can be written as

$$\begin{aligned} S_{\text {additional }} & =\left[\frac{k_{\mathrm{e}}}{4} \ln \left(\frac{1}{R_{2}}\right)\right. \\ & \left.\times \frac{e^{g W_{1}}-1}{g W_{1}}\left(1+R_{\mathrm{a}} e^{g W_{1}}\right)\left(1-R_{\mathrm{a}}\right)\right] S_{\text {control }} \end{aligned}$$

where the parameter $k_{\mathrm{e}}$ has been introduced into the model as a user-definable curve fitting parameter to account for process dependent variations and variations in device geometries, and to compensate for any simplify-
ing assumptions made in the derivation. Finally, the additional photon density is multiplied by the area of the overlap region and divided by the area of the output laser to average this contribution over the entire cavity.

$$S_{\text {extra }}=S_{\text {additional }} \frac{W_{1} W_{2}}{W_{2} L_{1}}$$

Thus, the extra photon density in the overlap region contributed by the control laser can be expressed as

$$S_{\text {extra }}=K S_{\text {control }}$$

where $K$ is referred to as the overlap or $K$ factor and is represented by

$$\begin{aligned} K= & \frac{k_{\mathrm{e}}}{4} \frac{W_{2}}{L_{1}} \ln \left(\frac{1}{R_{2}}\right) \frac{e^{g W_{1}}-1}{g W_{1}} \\ & \times\left(1+R_{\mathrm{a}} e^{g W_{1}}\right)\left(1-R_{\mathrm{a}}\right) \end{aligned}$$

It is important to realise that the overlap factor is highly geometry-dependent. The saturable gain and saturable absorption models assume simple rectangular cavities. For different geometries, an entirely different overlap factor must be derived. Alternatively, the user scalable parameter $k_{\mathrm{e}}$ can be adjusted to account for any new or different geometries. Although a full derivation of the new geometry would be the most accurate method, adjustment of $k_{\mathrm{e}}$ would be the most efficient for modelling purposes, resulting in the fastest turnaround time.